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New exact special solutions with solitary patterns for Boussinesq-like \(B(m,n)\) equations with fully nonlinear dispersion. (English) Zbl 1092.35021
The Boussinesq-like equations with fully nonlinear dispersion, \(B(m,n)\) equations \[ u_{tt}+(u^m)_{xx}-(u^n)_{xxxx}=0\tag{*} \] are considered. A new method different from the Adomian decomposition method to seek explicit and exact special solutions with solitary patterns for (*) are presented. In Section 2 the scheme of new “extend sinh-cosh” method to obtain multiple exact solutions for the nonlinear equation \(P(u,u_x,u_t,u_{xx},\dotsc) =0\) is given. In Section 3 two special cases, \(B(2,2)\) and \(B(3,3)\) are chosen to illustrate the concrete scheme of the general method. In particular, for \(B(2,2)\) new solution of the form \(u(x,t)=-2\omega^2/(3k^2)+a\exp\left(1/2(x+t\omega/k+ \xi_0)\right)\) and for \(B(3,3)\) the known solution obtained in [Y. Zhu, Chaos Solitons Fractals 22, No. 1, 213–220 (2004; Zbl 1062.35125)] are derived. In Section 4 a general formula for exact solutions of the \(B(n,n)\) equations, where \(n\) is a real number such that \(n\not=\pm1, n\not=0\) is established. For \(n<1\), (\(n\not=0,-1\)) the solitary wave solutions and the singular travelling wave solutions are obtained.

MSC:
35C05 Solutions to PDEs in closed form
35Q53 KdV equations (Korteweg-de Vries equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
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